I am a graduate student and want to study the theory of currents. What is a good reference for a beginner? I should be familiar with the theory of distributions or generalized functions on $\mathbb R^n$.


The theory of currents is a part of the geometric measure theory. Unfortunately, Federer made the subject completely inaccessible after he wrote his famous monograph:

H. Federer, Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York 1969.

The problem is that the book contains `everything' (well, almost) and it is unreadable. After this book was published, people did not dare to write other books on the topic and only the bravest hearts dared to read Federer's Bible.

In my opinion the first accessible book on the subject is

L. Simon, Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis, Australian National University, 3. Australian National University, Centre for Mathematical Analysis, Canberra, 1983.

You can find it as a pdf file in the internet. Note that this book was written 14 years after Federer's book and there was nothing in between.

I would also suggest:

F. Lin, X. Yang, Geometric measure theory—an introduction. Advanced Mathematics (Beijing/Boston), 1. Science Press Beijing, Beijing; International Press, Boston, MA, 2002.

I haven't read it, but it looks relatively elementary (relatively, because by no means the subject is elementary).

The last, but not least is

F. Morgan, Geometric measure theory. A beginner's guide. Fifth edition. Illustrated by James F. Bredt. Elsevier/Academic Press, Amsterdam, 2016.

You will not learn anything form that book as it does not have detailed proofs, but you can read it rather quickly and after that you will have an idea about what it is all about.

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  • $\begingroup$ Wikipedia also lists Fomenko, Anatoly T. (1990), Variational Principles in Topology (Multidimensional Minimal Surface Theory), Mathematics and its Applications (Book 42), Springer, Kluwer Academic Publishers, ISBN 978-0792302308; and Mattila, Pertti (1999), Geometry of Sets and Measures in Euclidean Spaces, London: Cambridge University Press, с. 356, ISBN 978-0-521-65595-8; but I do not know if they are good. $\endgroup$ – Oleg Lobachev Sep 21 at 21:01
  • $\begingroup$ @OlegLobachev I don't know Fomenko's book, but Mattila's book does not discuss currents. $\endgroup$ – Piotr Hajlasz Sep 22 at 2:03
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    $\begingroup$ One important note about Morgan's book is that at least its first half is meant as a kind of companion to Federer's magnum opus, i.e. they share the same notation and for many theorems Morgan will only give the important ideas and then point to the precise corresponding section of Federer's book, where all the technical details are carried out. Also, without wanting to spoil too much, the book is worth at least a look for the illustrations alone. $\endgroup$ – mlk Sep 22 at 16:18

A beginner friendly introduction can be found in chapter 7 of the book "Geometric Integration Theory " by Krantz and Parks. It is from 2008 and written in a modern and clear style and it starts nearly from "zero".

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